psubclsubN - Mathbox for Norm Megill

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Theorem psubclsubN 39942

Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
psubclsub.s	⊢ 𝑆 = (PSubSp‘𝐾)
psubclsub.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
psubclsubN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

**Proof of Theorem psubclsubN**
Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ (⊥_𝑃‘𝐾) = (⊥_𝑃‘𝐾)
2		psubclsub.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3	1, 2	psubcli2N 39941	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋)
4		eqid 2737	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	4, 1, 2	psubcliN 39940	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋))
6	5	simpld 494	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7		psubclsub.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 7, 1	polsubN 39909	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
9	6, 8	syldan 591	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
10	4, 7	psubssat 39756	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
11	9, 10	syldan 591	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
12	4, 7, 1	polsubN 39909	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
13	11, 12	syldan 591	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
14	3, 13	eqeltrrd 2842	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 Atomscatm 39264 HLchlt 39351 PSubSpcpsubsp 39498 ⊥_𝑃cpolN 39904 PSubClcpscN 39936

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18340 df-poset 18359 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-psubsp 39505 df-pmap 39506 df-polarityN 39905 df-psubclN 39937

This theorem is referenced by: pclfinclN 39952

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